#140527

order	512
nilpotency class	4
isoclinism	isoclinic class of #11677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(2,4)(5,6,7,9,8,11,12,10)(13,14)(15,23,26,36)(16,28)(17,20)(18,32)(19,25,33,29)(21,24)(22,31)(27,42,41,37)(30,43)(34,38,44,40)(35,39);(6,9,11,10)(13,15,24,38,31,41,16,29,17,26,39,40,18,27,30,25)(14,19,28,42,32,44,21,36,20,33,43,37,22,34,35,23);(1,2,3,4)(14,20)(19,33)(21,35)(22,32)(23,36)(28,43)(34,44)(37,42)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=ei,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47723);